Please, Please help me out here. I've got a test tomorrow.?

Question

So I'm unable to get past these three problems at the end of my review. Please answer and explain them. First correct answer will surely get 10 points.
1. g(x)= x+b, x<0; -cos(x), x>= 0. Now is there a value that will make g(x) continuous at x=0. Is there a value for g(x) that makes it diffrentiable at 0. why/why not? give reasons.
2. y=lnx and y=e^x are inverse functions. the graph of y=lnx looks like it may be approaching a horizontal asymptote. write an argument based on the graph of e^x to explain why it does not.
3. find the points on the graph of y=sec x, 0= Thanks a lot guys.

Answer 1

1) If you're looking for the value of b, then I'd say it's -1, since at 0, -cos(x) is equal to -1. As for the second part, it would not be differentiable into another function since the graph forms a corner at that point. In order for a function to be differentiable, it has to be continuous, can't be undefined, and can't have corners (like absolute value).
2) Even though the graph of ln(x) does look like it's flattening out as it approaches infinity, it does not have a horizontal asymptote. The domain and range of a function correspond to the range and domain of its inverse, respectively. e^x 's domain reaches from negative infinity to positive infinity, with no gaps in between. And since e^x 's domain corresponds to ln(x) 's range, this means that the range of ln(x) goes from negative infinity to positive infinity as well, also with no gaps in between. This precludes any possibility of a horizontal asymptote, as that would produce a gap in a function's range.
3) First of all, 9y-6x=12 can be converted to y=4/3+(2/3)x. So the slope of 9y-6x=12 is 2/3, and that's what we need. Now the slope of the tangent of y=sec(x) can be found from its derivative: y'=tan(x)sec(x). The place where the Y value on this graph is equal to 2/3 within a range of 0 to 2pi would be the answer you seek. But I'm guessing you already know all that and the real trouble is with the exact values part. And I'm sorry to say that I can't help you with that. The best I can give you is 0.52359877. I know it's not much help, but maybe you can figure out is it has any discernible relationship to pi or some exponents.
Good luck on your test.

Answer 2

2+2=5

Answer 3

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Answer 4

1)yes there is.
g(0)=-cos0=-1
g(0-)=0+b=b
g(0+)=-cos0=-1
g(0)=lim g(0-)=lim g(0+)
-1=b=-1
Hence b=-1


3)y=sec x, 0= slope=6/9=2/3

y=1/cosx
y'=sinx/cos^2 x

sinx/(1-sin^2x)=2/3
3sinx=2-2sin^2 x
let sinx =t

2t^2+3t-2=0
t1=-3+5/4=1/2
t2=-2 does not satisfy

hence sinx=1/2
x=30,150
or x=pi/6, 5pi/6

Answer 5

Oh damn! i really suck in math. I suggest you make some cheating list dude.

Answer 6

Did I mention,,,I love nerds! Umm,,,good luck with that!

Answer 7

a math? sry not on that chap yet